An idiom meaning someone's doing something useless and has no result at the end

There are a few idioms like that. One is:

You're just beating your head against the wall.

another is

Like trying to squeeze blood from a turnip.

and another is:

It's like trying to herd cats.

They each are used is slightly different situations. To be most analogous to your expression it sounds like I'd use the first one: Trying to convince him is like beating your head against the wall

Sisyphean labor. "Sisyphean" means endless and unavailing, as labor or a task. Comes from Greek mythology, Sisyphus was made to roll a giant boulder up a mountain as punishment by the gods, but the boulder would always roll down before he could get it to the top, forcing him to begin the pointless exercise anew.

There are very many idioms that you could use for a Sisyphean task. Take your pick.

I think the most generally applicable would be "pushing water uphill with a rake".
If you want to say that you're doing the right thing, but you're doing it too late to have any effect, you'd say "shutting the stable door after the horse has bolted". This is sometimes known as "stable door shutting" for short.
If you want to say that there are too many unpredictable factors to make your task possible, you'd talk about "herding cats".
If you want to say that you're trying unsuccessfully to get someone to do something or give you something, this is "trying to get blood out of a stone".

Plus all the other excellent suggestions in the other answers here.

To add to the other good examples: